270 ANALYTICAL MECHANICS 3. While passing through the tail of a comet an amount of dust of mass m settles uniformly upon the surface of the earth. Find the consequent change in the length of the day. 4. In the preceding problem find the torque due to the addition of mass. Suppose the passage to take n days and the rate at which mass is acquired to be constant. 6. A particle revolves, on a smooth horizontal plane, about a peg, to which"it is attached by means of a string of negligible mass. The string winds around the peg as the particle rotates. Discuss the motion of the particle. 6. A mouse is made to run around the edge of a horizontal circular table which is free to rotate about a vertical axis through the center. Find the velocity of the mouse relative to the table which will give the latter 20 revolutions per minute? The table weighs 2 pounds and has a diameter of 18 inches; the mouse weighs 5 ounces. 7. In the preceding problem find the velocity of the mouse with respect to the ground. 8. A cylindrical vessel of radius a is filled with a liquid, closed tight, and made to rotate with a constant angular velocity 6;0 about its geometrical axis, which is vertical. Suppose the frictiorial forces between the .inner surface of the vessel and the liquid and between the molecules of the liquid to be small, yet enough to transmit the motion to the liquid if the rotation is kept up for a long time. After each particle of water attains an angular velocity about the axis given by the relation co = co0r the torque which kept the angular velocity constant is stopped and the liquid is suddenly frozen. What will be the angular velocity of the system if (a) The mass of the vessel is negligible. (b) The mass is not negligible but the thickness is. Take the ends into account. (c) Neither the mass nor the thickness of the cylinder is negligible. Do not take the ends into account. (d) In (c) take the ends into account. 9. In the preceding problem suppose the distribution of the angular velocity of the liquid about the axis just before it is frozen to be givea aŚ r by the relation w = co0e T , where r is the, distance from the axis.